Abstract: The paper is devoted to homogenization of two-phase incompressibleviscoelastic flows with disordered microstructure. We study two cases. In thefirst case, both phases are modeled as Kelvin-Voight viscoelastic materials. Inthe second case, one phase is a Kelvin-Voight material, and the other is aviscous Newtonian fluid. The microscale system contains the conservation ofmass and balance of momentum equations. The inertial terms in the momentumequation incorporate the actual interface advected by the flow. In theconstitutive equations, a frozen interface is employed. The interface geometryis arbitrary: we do not assume periodicity, statistical homogeneity or scaleseparation. The problem is homogenized using G-convergence and oscillating testfunctions. Since the microscale system is not parabolic, previously knownconstructions of the test functions do not work here. The test functionsdeveloped in the paper are non-local in time and satisfy divergence-freeconstraint exactly. The latter feature enables us to avoid working withpressure directly. We show that the effective medium is a single phaseviscoelastic material that is not necessarily of Kelvin-Voight type. Theeffective constitutive equation contains a long memory viscoelastic term, aswell as instantaneous elastic and viscous terms.